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Groups with many roots | ||
| International Journal of Group Theory | ||
| مقاله 4، دوره 9، شماره 4، اسفند 2020، صفحه 261-276 اصل مقاله (254.71 K) | ||
| نوع مقاله: Proceedings of the conference "Engel conditions in groups" - Bath - UK - 2019 | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2020.119870.1582 | ||
| نویسندگان | ||
| Sarah Hart* 1؛ Daniel McVeagh2 | ||
| 1Birbeck, University of London | ||
| 2Department of Economics, Mathematics and Statistics, Birkbeck, University of London | ||
| چکیده | ||
| Given a prime $p$, a finite group $G$ and a non-identity element $g$, what is the largest number of $p^{th}$ roots $g$ can have? We write $ϱ_p(G)$, or just $ϱ_p$, for the maximum value of $\frac{1}{|G|}|\{x \in G: x^p=g\}|$, where $g$ ranges over the non-identity elements of $G$. This paper studies groups for which $ϱ_p$ is large. If there is an element $g$ of $G$ with more $p^{th}$ roots than the identity, then we show $ϱ_p(G) \leq ϱ_p(P)$, where $P$ is any Sylow $p$-subgroup of $G$, meaning that we can often reduce to the case where $G$ is a $p$-group. We show that if $G$ is a regular $p$-group, then $ϱ_p(G) \leq \frac{1}{p}$, while if $G$ is a $p$-group of maximal class, then $ϱ_p(G) \leq \frac{1}{p} + \frac{1}{p^2}$ (both these bounds are sharp). We classify the groups with high values of $ϱ_2$, and give partial results on groups with high values of $ϱ_3$. | ||
| کلیدواژهها | ||
| $pth$ roots؛ square roots؛ cube roots | ||
| مراجع | ||
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[1] Y. Berkovich, Groups of prime power order, Vol. 1, Walter de Gruyter, Berlin (2008).
[2] Y. Berkovich, On the number of solutions of the equation xpk= a in a finite p-group, Proc. American Math. Soc., 116 (1992) 585–590. [3] N. Blackburn, Note on a paper of Berkovich, J. Algebra, 24 (1973) 323–334.
[4] G. A. Fern´andez-Alcober, An introduction to finite p-groups: regular p-groups and groups of maximal class, Mat. Contemp., 20 (2001) 155–226. [5] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.10.2; 2019. https://www.gap-system. org. [6] B. Huppert, Endliche Gruppen I, Grundlehren der Mathematischen Wissenschaften, 134, Springer-Verlag, Berlin, (1967). [7] T. J. Laffey, The Number of Solutions of xp = 1 in a Finite Group, Mathematical Proceedings of the Cambridge Philosophical Society, 80 (1976) 229–31. [8] T. J. Laffey, The Number of Solutions of x3 = 1 in a 3-group, Math. Zeitschrift., 149 (1976) 43–45.
[9] T. Y. Lam, On the number of solutions of xpk= a in a p-group, Illinois J. Math., 32 (1988) 575–583.
[10] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language. Computational algebra and number theory (London, 1993), J. Symbolic Comput., 24 (1997) 235–265. [11] C. T. C. Wall, On groups consisting mostly of involutions, Proc. Camb. Phil. Soc., 67 (1970) 251–262. | ||
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